Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 221 / 3 / iv / Solution

Let $M$ be a Markov blanket of the treatment $A$ inside $(A,X)$, and write the remaining adjustment variables as $N=X\setminus M$. By definition,The sufficiency of $X=(M,N)$ givesApply the contraction axiom for conditional independence with first variable $A$, second variable $N$, third variable $Y(a)$, and conditioning variable $M$. It givesThe decomposition axiom for conditional independence then yields $A\perp Y(a)\mid M$. Hence every Markov blanket of $A$ in $(A,X)$ is itself a sufficient adjustment set.